On the other hand, and aimed to also uniquely identify the edges (resp. edges and vertices) of a graph, the notion of edge (resp. mixed) resolving sets were first presented in [1]} (resp. [2]}) as follows. Two elements \(x,y\in V(G)\cup E(G)\)  are resolved by a vertex \(v\in V(G)\)  if \(d_G(x,v)\ne d_G(y,v)\) . Here, if \(x=ww^{\prime }\)  is an edge, then \(d_G(x,v)=\min \lbrace d_G(w,v), d_G(w^{\prime },v)\rbrace \) . A set of vertices \(S\subset V(G)\)  is said to be an edge resolving set for \(G\)  if every two edges \(e,f\in E(G)\)  are resolved by a vertex of \(S\) . Moreover, the set \(S\)  is called a mixed resolving set for \(G\)  if any two elements (vertices or edges) \(x,y\in V(G)\cup E(G)\)  of the graph are resolved by a vertex of \(S\) . An edge (resp. mixed) resolving set of the smallest possible cardinality is called an edge (resp. mixed) metric basis, and its cardinality the edge (resp. mixed) metric dimension, denoted by \({\rm edim}(G)\)  (resp. \({\rm mdim}(G)\) ). Recent studies on the edge and mixed metric dimensions of graphs are for instance [3]}, [4]} and [5]}, [6]}, respectively.
